Elementary math in elementary school: the effect of interference on learning the multiplication table

Memorizing the multiplication table is a major challenge for elementary school students: there are many facts to memorize, and they are often similar to each other, which creates interference in memory. Here, we examined whether learning would improve if the degree of interference is reduced, and which memory processes are responsible for this improvement. In a series of 16 short training sessions over 4 weeks, first-grade children learned 16 multiplication facts—4 facts per week. In 2 weeks the facts were dissimilar from each other (low interference), and in 2 control weeks the facts were similar (high interference). Learning in the low-similarity, low-interference weeks was better than in the high-similarity weeks. Critically, this similarity effect originated in the specific learning context, i.e., the grouping of facts to weeks, and could not be explained as an intrinsic advantage of certain facts over others. Moreover, the interference arose from the similarity between facts in a given week, not from the similarity to previously learned facts. Similarity affected long-term memory—its effect persisted 7 weeks after training has ended; and it operated on long-term memory directly, not via the mediation of working memory. Pedagogically, the effectiveness of the low-interference training method, which is dramatically different from currently used pedagogical methods, may pave the way to enhancing how we teach the multiplication table in school. Supplementary Information The online version contains supplementary material available at 10.1186/s41235-022-00451-0.

. The sets of facts trained in each week for each participant. Grey background indicates high-similarity sets. The stimuli are also provided in CSV format.

Participant
Week 1 Week 2 Week 3 Week 4     Pre-existing multiplication knowledge: the child answered correctly to at least one of the 16 facts to be learned in the experiment. "Correct" was defined as providing the correct answer to a particular fact at least twice (out of the 3 attempts).

Participant exclusion
 The child decided to quit.
 Not following the experiment rules: the parents of one participant provided help by teaching multiplication strategies, in contrast to the experiment instructions.
 Not cooperative means that according to the experimenter's best judgment, the child showed low motivation and was uninterested in the experiment to the degree that may impair performance.
 Inattentiveness means that according to the experimenter's best judgment, the participant was extremely inattentive, to the degree of being unable to focus on the task effectively.
 Repetition errors is an objective criterion for inattentiveness. It means that the participant made more than 15 repetition errors in at least one training day (Fig. S1). A "repetition error" refers to the first stage of each training round, in which the participant repeated the facts said by experimenter. Note that this criterion does not confound with the child's performance, and critically it does not confound with the effect of similarity, because the criterion is based on the children's ability to repeat the facts said by experimenter in the first part of each training round, whereas our measures of learning progress were based on the children's retrieval of facts in the second part of each training round.
The 15-errors threshold was not exceeded by any of the participants who remained in the study, except one participant (#25) in the first training day.  S1. The total number of repetition errors, i.e., repeating incorrectly a fact said by the experimenter, in each training day. Each line represents one participant, the legend shows participant numbers. Color codes: green = participants in the study; pink = excluded for being uncooperative; red/orange = excluded for inattentiveness.

Participant exclusion in Experiment 2
This data refers to the control experiment in which children performed only the forced-choice test, with no preceding learning. We excluded children with stereotypical responses -i.e., children who consistently chose the second alternative presented to them (no child consistently chose the first alternative). Table S4 shows the rate of trials in which each child chose the second alternative.

Deviations from the experiment protocol
In a small number of cases, the experiment slightly deviated from the standard training protocol. We hereby detail these cases and how they were addressed:  Learned exercise. Participant #19 started learning the exercise 36 in class during the study period. This specific exercise was therefore excluded from all analyses for that participant.
 Inattentive participant. One participant (#16) was extremely inattentive in two sessions in the 4 th training week. To compensate for this, we added a 5 th training day for that participant during that particular week.
 Human error. The experiment protocol dictated that when asking the children to retrieve the multiplication facts during the training rounds, the experimenter first asked the child to say any fact they remembered; then, if the child completely failed to mention some facts, the experimenter presented that fact and asked what the answer was. In very few cases, due to human error, the experimenters did the first part of the protocol, but did not follow to the second part -i.e., they did not present the facts that the child completely forgot. This was the case for 5 participants (#1, #4, #6, #12, #16), in the first 3 days of the first training week (for participant #1, only in the first day of the first week). To compensate for this, we added a 5 th training day to these children during the first week.
We verified that this minor breach of protocol did not affect the results. First, of the 5 children, the first week included high-similarity facts for 3 children (#1, #6, #16) and low-similarity facts for the other two. Thus, the experimenter mistake was equally distributed across lowsimilarity and high-similarity sets. Second, as Figure 2a clearly shows, the mistake did not affect the per-child end-of-week results.
In the analyses that examined the performance separately for each day, we considered only the first four days of each week and ignored the fifth training day, if it existed.

Pre-experiment addition and subtraction test
In this test, ran in the first day of the study (in week 1), each child solved 10 addition and 7 subtraction exercises. The average accuracy was 71.2% in the addition exercises and 53.8% in the subtraction exercises. This difference is significant (t(16) = 3.35, one-tailed p = .002), however, note that the additions and subtractions were not balanced. Table S5 shows the average accuracy for each exercise. Table S6 shows the average accuracy for each participant; unsurprisingly, there was much variance between participants. The full per-exercise, per-participant information is available in the "tests.csv" file in the supplementary data repository.

Fig. S2.
In the second round of the end-of-week tests, the effect of similarity could still be observed. The effect size was similar to the first round, but with lower significance levels.

Detailed results of linear mixed models
The following tables include the full details of each logistic linear mixed model described in the main text. In all these models, the dependent variable was the accuracy in each fact.